The discipline of applying mathematical analysis to complex decision problems in operations: allocating scarce resources, scheduling, routing, inventory, and the design of service and production systems. Drawing on mathematical programming, stochastic modeling, queueing, simulation, and game-theoretic reasoning, it seeks policies that perform provably well in systems shaped by constraints, congestion, and uncertainty.
Missions
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Captain: Shuze Chen
Bandit Algorithms I: Concentration of MeasureTextbook
How quickly does the empirical mean of independent random variables concentrate around the true mean? This question is the analytic engine of the entire theory of stochastic bandits: every optimistic algorithm (Explore-Then-Commit, UCB and its relatives) is calibrated by a tail bound on the sample mean. This mission formalizes the subgaussian framework of Chapter 5 of Lattimore–Szepesvári's *Bandit Algorithms*: a random variable $X$ is $\sigma$-subgaussian when $\mathbb{E}[e^{\lambda X}] \le e^{\lambda^2\sigma^2/2}$ for all $\lambda$, and the Cramér–Chernoff method converts this moment-generating-function control into the exponential tail $\mathbb{P}(X \ge \varepsilon) \le e^{-\varepsilon^2/(2\sigma^2)}$. The goal theorem is the Hoeffding-type bound: the sample mean of $n$ independent $\sigma$-subgaussian deviations exceeds the true mean by $\varepsilon$ with probability at most $\exp(-n\varepsilon^2/(2\sigma^2))$, together with its confidence form $\mathbb{P}\big(\hat\mu + \sqrt{2\sigma^2\log(1/\delta)/n} \le \mu\big) \le \delta$ — the exact bound every UCB index is built from. These few lines of analysis are cited by every regret bound in the series.
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Captain: tianyipeng
Markov Entanglement: Decomposition Error via Agent-wise TV DistanceResearch Paper
Multi-agent reinforcement learning approximates a global value function by summing per-agent local value functions learned independently — a trick that works surprisingly well in practice (ride-hailing dispatch, restless bandits) but had no general theoretical justification. Chen and Peng (arXiv:2506.02385) explain why: they define a *Markov entanglement* measure for the joint transition dynamics of a multi-agent MDP, directly analogous to quantum entanglement of a two-party state, and show it controls exactly how much error this value-decomposition trick incurs. This mission formalizes their sharpest quantitative bound (Theorem 4): the error of decomposing the global Q-function into per-agent local Q-functions is controlled, entrywise, by the agent-wise total-variation measure of Markov entanglement.
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Captain: qm2204
Buying to Bundle: Asymptotic Optimality of Surrogate BundlingResearch Paper
A platform sourcing items from monopolistic sellers with private quality cannot tractably maximize its true profit: the bundle revenue $Rev(v_S)$ is neither monotone, submodular, supermodular, subadditive, nor superadditive. Theorem 4.6 of *Buying to Bundle: Optimal Sourcing from Monopolistic Sellers* shows that the simple surrogate threshold mechanism — maximize the linearized objective $\varpi(x)=N\,E[x(\mu)(\mu-\varphi(\mu))]$ — is profit-optimal up to a $1+O(N^{-1/3})$ factor in large markets. Prove it: Bernoulli concentration for the bundle quality plus sub-exponential control of the dispersion gap $|Rev(v)-E[v]|$ (Lemma 4.5).
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Markov Entanglement: Index Policies for Restless Bandits are Asymptotically SeparableResearch Paper
Restless multi-armed bandits are the standard model for allocating a scarce resource across many independently-evolving agents: N arms, each a small Markov chain, and a budget that lets you activate only a fixed fraction of them at each step. The joint problem is PSPACE-hard, so practice runs on *index policies* — score each arm by a priority index computed from its own local state, then activate the top ones until the budget runs out — and evaluates them by *value decomposition*: approximate the joint Q-function by a sum of per-arm local Q-functions, each computed from a single arm's chain. The decomposition is used everywhere from Whittle-index heuristics to modern multi-agent RL, and it is used without an error bound.
Chen and Peng (arXiv:2506.02385) supply one. Their companion mission established the general principle: the value decomposition error of a multi-agent chain is controlled by its *measure of Markov entanglement*, the distance from the chain's transition matrix to the nearest separable one. This mission carries that principle to the restless-bandit setting and proves that index policies are asymptotically separable — their entanglement decays like 1/sqrt(N), so the decomposition error is sublinear in N while the joint Q-function itself is of order N. The relative error vanishes as the system grows, which is exactly why the practice works.
The argument runs through the mean-field limit. Because the arms are homogeneous, the only thing that matters about a joint state is its *configuration*: the fraction of arms in each local state. Under an index policy the configuration evolves by a map that does not depend on N at all, and under two standard technical conditions — a uniform global attractor property and non-degeneracy — that map has a unique attracting fixed point m*. The chain of reasoning is: policy entanglement is bounded by how far the realised policy sits from the mean-field limiting policy (Proposition 1); that distance is bounded by the configuration's deviation from m* (Lemma 2/8); and the deviation concentrates at rate 1/sqrt(N) by a concentration-plus-local-stability argument adapted from Gast, Gaujal and Yan. The concentration and stability inputs (Lemmas 9, 10, 11) are results of Gast et al. and are formalized here as well, so the mission stands on its own.
The mission also formalizes the mean-field map on the whole simplex and checks it against the N-agent characterisation, which is what makes the piecewise-affine and stability analysis expressible at all.
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Captain: wenxinzhang
Vector Space Methods IX: Global Lagrange DualityTextbook
## Motivation
Many convex programs impose inequalities valued in a vector space: componentwise inequalities, positive-semidefinite constraints, and families of ordered resource constraints are all instances of one cone order. Chapter 8 of David G. Luenberger's [*Optimization by Vector Space Methods*](https://openlibrary.org/books/OL7612943M/Optimization_by_Vector_Space_Methods) develops a global theory for this setting. A perturbation of the constraint produces a convex value function, continuous linear functionals positive on the ordering cone become Lagrange multipliers, and a strict-feasibility condition yields an attained dual optimum. This mission formalizes the progression in §§8.2–8.6, culminating in the book's Lagrange Duality Theorem.
## Setting
Let $X$ and $Z$ be real normed spaces, let $\Omega\subseteq X$ be a nonempty convex set, and let $P\subseteq Z$ be a **convex cone**. The cone induces the relation
$$
z_1\le_P z_2\quad\Longleftrightarrow\quad z_2-z_1\in P.
$$
A continuous linear functional $z^*\in Z^*$ is **dual-positive** when $z^*(p)\ge0$ for every $p\in P$. A map $G:X\to Z$ is **cone-convex on $\Omega$** when its value at a convex combination is below the corresponding convex combination of its values in this cone order. The primal program is
$$
\mu=\inf\{f(x):x\in\Omega,\ G(x)\le_P0\},
$$
where $f$ is real-valued and convex on $\Omega$.
For a multiplier $z^*$, the **Lagrangian** and its possibly infinite dual value are
$$
L(x,z^*)=f(x)+z^*(G(x)),\qquad
\phi(z^*)=\inf_{x\in\Omega}L(x,z^*).
$$
The perturbed primal value $\omega(z)$ replaces the zero right-hand side by $G(x)\le_P z$. Lean represents $\omega$ and $\phi$ in `EReal`, so infeasible perturbations have value $+\infty$ and objectives unbounded below can have value $-\infty$ without arbitrary defaults.
## Formalization targets
### Main goal: Lagrange duality
Assume $P$ has nonempty interior, the primal value $\mu$ is finite, and there is a **strictly feasible point** $x_s\in\Omega$ with
$$
-G(x_s)\in\operatorname{int}P.
$$
Prove that a dual-positive $z_0^*$ exists and attains
$$
\mu=\phi(z_0^*)=
\max_{z^*\ \text{dual-positive}}\phi(z^*).
$$
If $x_0$ attains the primal infimum, also prove complementarity $z_0^*(G(x_0))=0$ and that $x_0$ minimizes $L(\,·\,,z_0^*)$ over $\Omega$.
### Milestones
Five source milestones delimit the reusable theory. A closed convex cone is recovered from all dual-positive inequalities (§8.2, Proposition 1). The finite-height epigraph of the extended perturbation value is convex, and that value is antitone in the cone order (§8.3, Propositions 1–2). A Lagrangian saddle point is sufficient for primal feasibility and optimality when the cone is closed (§8.4, Theorem 2). Finally, multipliers for two perturbed right-hand sides bound the change in optimal objective value from both sides (§8.5, Theorem 1). The root then states §8.6, Theorem 1 rather than duplicating the equivalent multiplier theorem from §8.3.
## Significance
The capstone provides both equality of optimal values and an attained multiplier. It applies to a single vector inequality, so finite systems of scalar inequalities and matrix-cone constraints fit the same statement once their ordering cones are supplied. Complementarity and Lagrangian minimization turn a primal optimizer and multiplier into a certificate. The sensitivity milestone additionally gives quantitative information about how the optimum changes when the constraint right-hand side moves.
Formalization produces a reusable cone-order layer independent of coordinate choices. `coneLE`, `dualPositive`, and `ConeConvexOn` can support later Kuhn–Tucker, vector optimization, and conic programming developments. The `EReal` value functions preserve infeasibility and unboundedness, two cases that a real-valued `sInf` encoding would collapse. This is a formalization mission for a classical theorem, not a claim that the underlying duality result is open.
## Difficulty
The theorem's strict-feasibility condition is load-bearing. Feasibility $-G(x)\in P$ cannot replace interior feasibility, and nonempty interior of $P$ alone does not supply a Slater point. Equality constraints also cannot be converted into pairs of inequalities while retaining strict feasibility; Luenberger explicitly warns about this after the theorem.
The cone assumptions differ across milestones. The main strong-duality theorem does not require $P$ to be closed or pointed, whereas the bipolar and saddle-sufficiency statements require closedness. Using Mathlib's stronger `ProperCone` everywhere would silently add both topological and order hypotheses and shrink the theorem. Another tempting simplification is to make both value functions real. That loses the empty feasible set and unbounded dual subproblem, precisely the boundary cases used when comparing perturbations. The saddle inequalities must also have the correct orientation: the multiplier coordinate is maximized and the primal coordinate is minimized.
## Formalization scope
The mission uses `ConvexCone ℝ Z` with a custom induced relation; it deliberately does not assume a lattice order on $Z$. Multipliers are continuous linear maps $Z\to\mathbb R$. The root assumes a real finite optimum through `IsGLB` and a real witness $\mu$, while `lagrangeDualValue` and `perturbationValue` retain `EReal` codomains. The strict condition is written as membership of $-G(x_s)$ in `interior P`, exactly matching $G(x_s)<_P0$.
No finite-dimensionality, reflexivity, completeness, closedness, or pointedness is added to the root. Closedness appears only where the source uses cone separation to recover primal feasibility. The sensitivity item assumes the two candidate points are feasible, their multipliers are dual-positive and complementary, and each point minimizes its shifted Lagrangian; these hypotheses spell out “solutions and corresponding multipliers” without relying on informal terminology.
Contributions may formalize cone separation, perturbation-value geometry, saddle certificates, or strong duality. Finite-dimensional orthant and positive-semidefinite specializations are useful corollaries but do not replace the general goal. Local multiplier rules, equality constraints, differentiable Kuhn–Tucker conditions, and Chapter 9's local theory remain outside this mission.
## Selected references
- David G. Luenberger, *Optimization by Vector Space Methods*, John Wiley & Sons, 1969, Chapter 8, §§8.2–8.6, pp. 214–225. [Open Library record](https://openlibrary.org/books/OL7612943M/Optimization_by_Vector_Space_Methods)
- Stephen Boyd and Lieven Vandenberghe, *Convex Optimization*, Cambridge University Press, 2004, Chapter 5. [Official book page](https://web.stanford.edu/~boyd/cvxbook/)
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Captain: Shuze Chen
Treatment Locality in A/B TestingResearch Paper
Modern A/B tests must infer *lifetime* treatment effects — e.g. customer lifetime value under a new feature — from short-horizon experiment data. Chen, Simchi-Levi and Wang (arXiv:2407.19618) model the experiment as a Markov decision process and exploit a structural fact of many practical interventions: the treatment is *local*, modifying the system at a single crucial state only. This mission formalizes the core asymptotic theory of the paper: for **any** differentiable estimator built from the experiment's transition and reward statistics, *information sharing* — pooling across test arms the samples collected away from the treated state — keeps the estimator asymptotically normal with the same asymptotic bias and never increases its asymptotic variance (Theorem 9), and is asymptotically efficient among unbiased estimators (Theorem 5). The route runs through a Markov chain central limit theorem with the asymptotic variance identified as the autocovariance series, and the linearization/delta method for functionals of chain statistics.
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Captain: Shuze Chen
Introduction to Linear Optimization VI: Farkas' Lemma and Separating HyperplanesTextbook
When is a system of linear constraints infeasible? Sections 4.6-4.7 of Bertsimas-Tsitsiklis answer with the archetypal theorem of the alternative. The capstone is Farkas' lemma (Theorem 4.6): for an $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, exactly one of the following holds — (a) some $x \ge 0$ satisfies $Ax = b$, or (b) some $p$ satisfies $p'A \ge 0'$ and $p'b < 0$; such a $p$ is a certificate of infeasibility, geometrically a hyperplane separating $b$ from the cone of the columns of $A$. The mission also carries the cone-membership restatement (Corollary 4.3), the inequality form (Theorem 4.7: every solution of $Ax \le b$ satisfies $c'x \le d$ iff some $p \ge 0$ has $p'A = c'$ and $p'b \le d$), and the application to asset pricing (Theorem 4.8: a market's prices admit no arbitrage iff there is a nonnegative state-price vector $q$ with $p_i = \sum_s q_s r_{si}$). The book proves Farkas' lemma from LP strong duality; Section 4.7 then reverses the arrow from first principles: every polyhedron is closed (Theorem 4.9), Weierstrass' theorem (Theorem 4.10, already in Mathlib), and the separating hyperplane theorem (Theorem 4.11: for nonempty closed convex $S$ and $x^* \notin S$ there exists $c$ with $c'x^* < c'x$ for all $x \in S$), from which Farkas' lemma — and hence the duality theorem itself — follows geometrically.
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Captain: Shuze Chen
Introduction to Linear Optimization V: Duality TheoryTextbook
Every linear programming problem has a shadow. To the primal $\min c'x$ we associate the dual $\max p'b$, whose variables price the primal constraints: one dual variable per primal constraint and one dual constraint per primal variable, with signs governed by the correspondence of Table 4.1. This mission formalizes §4.1–4.5 of Bertsimas–Tsitsiklis: the dual of a general-form linear program, the involution "the dual of the dual is the primal" (Theorem 4.1), and weak duality $p'b \le c'x$ for any primal-feasible $x$ and dual-feasible $p$ (Theorem 4.3) with its two corollaries — an unbounded primal forces an infeasible dual (Corollary 4.1), and feasible $x, p$ with $p'b = c'x$ are automatically both optimal (Corollary 4.2). The goal theorem is strong duality (Theorem 4.4): if a linear programming problem has an optimal solution, so does its dual, and the respective optimal costs are equal — proved in the book by running the simplex method with the lexicographic pivoting rule of Mission IV on a standard-form transform. The statement is deliberately the book's attainment form: by Table 4.2 the primal and the dual can be simultaneously infeasible (Example 4.5), so an unguarded equality of optimal values is false. The mission closes with complementary slackness (Theorem 4.5): feasible $x$ and $p$ are simultaneously optimal if and only if $p_i(a_i'x - b_i) = 0$ for all $i$ and $(c_j - p'A_j)x_j = 0$ for all $j$ — the certificate structure behind the dual simplex method and every LP optimality check.
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Captain: tianyipeng
Hefferon Linear Algebra I: Gauss's Method and the Solution SetTextbook
Chapter One of Jim Hefferon's *Linear Algebra* develops Gauss's method and asks what row reduction actually preserves. The answer arrives as the Linear Combination Lemma: row operations change the rows of a matrix but never the subspace those rows span, and that invariant is complete. The goal theorem is that completeness — two matrices are row equivalent exactly when they have the same row space — which is what makes reduced echelon form a genuine canonical form. The milestones are the two results the chapter builds on the way: that row operations leave a system's solution set alone, and that a solution set is always one particular solution translated by the solutions of the associated homogeneous system.
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Captain: Shuze Chen
Introduction to Linear Optimization II: Existence and Optimality of Extreme PointsTextbook
Where should one look for the optimum of a linear programming problem? Chapter 1 of Bertsimas–Tsitsiklis suggests that optima "tend to occur at corners" of the feasible polyhedron; §§2.5–2.6 turn this intuition into theorems. Not every polyhedron has a corner — a halfspace in $\mathbb{R}^n$ ($n > 1$) has none — and the exact dividing line is the presence of an infinite line: a nonempty polyhedron $$P = \{x \mid a_i'x \ge b_i,\ i = 1, \dots, m\}$$ has an extreme point if and only if it does not contain a line, if and only if $n$ of the vectors $a_1, \dots, a_m$ are linearly independent (Theorem 2.6). In particular every nonempty bounded polyhedron and every nonempty standard-form polyhedron has a basic feasible solution (Corollary 2.2). The capstone, Theorem 2.8, is the sharpest form of the corner principle: if $P$ has at least one extreme point, then for any cost vector $c$ either the optimal cost is $-\infty$, or there is an extreme point of $P$ that is optimal — existence of an optimal solution comes for free once the cost is bounded below. Its companion Theorem 2.7 places an optimal extreme point under the weaker assumption that an optimal solution exists, and Corollary 2.3 — the fundamental theorem of linear programming — concludes that every feasible LP either has optimal cost $-\infty$ or attains an optimal solution, in stark contrast with nonlinear problems such as minimizing $1/x$ over $x \ge 1$. These results license the extreme-point search that the simplex method (Mission IV) performs.
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Captain: Shuze Chen
Bandit Algorithms XV: Partial MonitoringTextbook
Bandit feedback is only one point on a spectrum: a learner might see more than its own loss (full information) or less (a spam filter never learns what happened to mail it deleted). Chapter 37 of Lattimore–Szepesvári studies finite adversarial games $G = (\mathcal{L}, \Phi)$ where the loss matrix and the feedback matrix are decoupled. The goal theorem is the celebrated classification theorem: every finite partial-monitoring game has minimax regret exactly $0$, $\Theta(\sqrt{n})$, $\Theta(n^{2/3})$ or $\Omega(n)$ — determined by two purely combinatorial conditions, global and local observability, on the game's neighbourhood structure. A single geometric dichotomy thus governs the price of information in every online decision problem with finite actions and feedback.
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Captain: Shuze Chen
Bandit Algorithms XII: Follow-the-Regularised-Leader and Mirror DescentTextbook
Beneath Exp3, Exp4 and their relatives lies one algorithm: minimize past losses plus a convex regularizer. Chapters 26–28 of Lattimore–Szepesvári develop this unifying view. For a Legendre potential $F$ with Bregman divergence $D_F$, both mirror descent and follow-the-regularised-leader satisfy the master bound $R_n(a) \le \frac{F(a) - F(a_1)}{\eta} + \frac{1}{\eta}\sum_t D_F(a_t, \tilde a_{t+1})$; the negentropy potential on the simplex recovers Exp3 exactly. The goal theorem is the payoff for adversarial *linear* bandits: FTRL on the unit ball with the self-concordant-flavoured potential $F(a) = -\log(1-\|a\|) - \|a\|$ achieves $R_n \le 2\sqrt{3nd\log n}$ — improving the $\sqrt{d}$ factor over the Exp3-style approach of Chapter 27 and matching the $\Omega(d\sqrt{n})$ lower bound of Mission XI up to logarithms.
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Captain: Shuze Chen
Bandit Algorithms XI: Lower Bounds for Stochastic Linear BanditsTextbook
Is the $d\sqrt{n}$ regret of LinUCB (Mission X) an artifact of the algorithm or a law of nature? Chapters 24–25 of Lattimore–Szepesvári prove it is essentially unimprovable. On the unit ball there is a parameter $\theta$ with $\|\theta\|_2^2 = d^2/(48n)$ forcing $R_n \ge \frac{d\sqrt{n}}{16\sqrt{3}}$ — the goal theorem — and the hypercube gives the same $\Omega(d\sqrt{n})$ rate. The asymptotic chapter is more striking still: for fixed finite action sets, the instance-optimal constant $c(\mathcal{A},\theta)$ is characterized by an allocation program, and *optimism itself is provably suboptimal* — LinUCB and Thompson sampling cannot achieve it, because exploration must sometimes deliberately play actions optimism would never touch. These lower bounds define the targets for the entire linear-bandit literature.
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Captain: Shuze Chen
Bandit Algorithms X: Stochastic Linear Bandits and LinUCBTextbook
When actions are feature vectors and the mean reward is linear — $X_t = \langle \theta_*, A_t\rangle + \eta_t$ — a bandit can generalize across arms: pulling one arm reveals information about all of them. Chapter 19 of Lattimore–Szepesvári carries the optimism principle into this setting: LinUCB (a.k.a. OFUL) plays the action maximizing $\max_{\theta\in\mathcal{C}_t}\langle\theta, a\rangle$ over the confidence ellipsoid $\mathcal{C}_t$ of Mission IX. The goal theorem: with probability $1-\delta$, $\hat R_n \le \sqrt{8n\beta_n \log\frac{\det V_n}{\det V_0}} \le \sqrt{8dn\beta_n\log\frac{d\lambda + nL^2}{d\lambda}}$ — regret $\tilde O(d\sqrt{n})$ independent of the number of actions. The combinatorial engine is the elliptical potential lemma, bounding how many times adaptively chosen directions can be surprising. Chapter 22's phased elimination with G-optimal design (Mission IX) sharpens this to $\tilde O(\sqrt{dn\log k})$ for finite action sets.
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Captain: Shuze Chen
Bandit Algorithms IX: Self-Normalized Concentration and Optimal DesignTextbook
Least-squares estimation from *adaptively* collected data is the statistical heart of linear bandits: the actions $A_t$ depend on past noise, so classical fixed-design theory does not apply. Chapter 20 of Lattimore–Szepesvári resolves this with the method of mixtures: the process $M_t(x) = \exp(\langle x, S_t\rangle - \frac{1}{2}\|x\|^2_{V_t(\lambda)})$ is a supermartingale, and integrating over a Gaussian mixture yields the self-normalized bound — the goal theorem — $\mathbb{P}\big(\exists t : \|S_t\|^2_{V_t(\lambda)^{-1}} \ge 2\log\frac{1}{\delta} + \log\frac{\det V_t(\lambda)}{\lambda^d}\big) \le \delta$, valid uniformly over all times. The resulting confidence ellipsoids for the regularized least-squares estimator (Abbasi-Yadkori et al.) calibrate every algorithm of Mission X. The mission also formalizes the Kiefer–Wolfowitz theorem of Chapter 21: G-optimal and D-optimal experimental designs coincide, with optimal value exactly $d$ — the classical equivalence theorem of optimal design theory.
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Bandit Algorithms VIII: Contextual Bandits and Exp4Textbook
Real decisions come with context: a news site chooses an article *for a particular user*. Competing with the single best arm is then meaningless; the right benchmark is the best mapping from contexts to arms, or more generally the best of $M$ expert policies. Chapter 18 of Lattimore–Szepesvári formalizes this via Exp4 — exponential weighting over experts, fed by the importance-weighted estimator of Mission V. The goal theorem: with learning rate $\eta = \sqrt{2\log(M)/(nk)}$, Exp4 satisfies $R_n \le \sqrt{2nk\log M}$ against the best of $M$ experts. Since $M$ enters only logarithmically, the learner can compete with exponentially large policy classes — the conceptual gateway from bandits to reinforcement learning with function approximation.
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Captain: wenxinzhang
Single-Server Queueing Convergence via Forward CouplingTextbook
Formalize sample-path stability for a continuous-time, unit-rate, infinite-buffer single-server queue. Starting from cumulative arriving service work, define the reflected transient workload, the workload constructed from the infinite past, long-run offered load, and two-time stationarity. Prove that subcritical load forces finite-time coupling and consequently that every finite initial workload converges in its two-time finite-dimensional distributions to the stationary workload law.
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Captain: wenxinzhang
Sample-Path Little's LawTextbook
Formalize sample-path Little's Law for deterministic continuous-time queueing trajectories, decomposed into area, sojourn, arrival-rate, boundary, and squeeze lemmas.
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Vector Space Methods II: Gauss–Markov EstimationTextbook
## Motivation
Chapter 4 of Luenberger's *Optimization by Vector Space Methods* (Wiley, 1969) develops linear least-squares estimation as an application of the Hilbert space projection theorem formalized in Mission I of this series. The chapter's centerpiece is the classical **Gauss–Markov theorem**: among all linear unbiased estimators of an unknown parameter vector from noisy linear measurements, the estimator $(W^\top Q^{-1} W)^{-1} W^\top Q^{-1} y$ has minimum variance — componentwise, not merely in trace. This result is foundational for statistics and econometrics, and its Hilbert-space derivation is the cleanest known.
## Setting
Measurements are modeled as $y = W\beta + \varepsilon$, where $y$ is an $m$-dimensional data vector, $W$ a known $m \times n$ matrix ($n < m$) with linearly independent columns, $\beta$ an unknown $n$-dimensional parameter vector, and $\varepsilon$ a random $m$-vector of measurement errors with $E\varepsilon = 0$ and covariance $E[\varepsilon\varepsilon^\top] = Q$, positive definite. A **linear estimate** is $\hat\beta = Ky$ for a constant $n \times m$ matrix $K$; it is **unbiased** when $E\hat\beta = \beta$ for every $\beta$, which holds iff $KW = I$. The optimality criterion is the error second moment $E\|\hat\beta - \beta\|^2$, and the book's key observation (p. 85) is that the problem splits into $n$ independent minimum norm problems, one per component, each solvable by the dual approximation theorem of Mission I.
Formally, randomness is carried by an abstract probability space: a measure space $(\Omega, \mu)$ with $\mu$ a probability measure, random vectors as functions $\Omega \to \mathbb{R}^m$ with explicit integrability hypotheses for all first and second moments, and $E[\cdot] = \int \cdot \, d\mu$.
## Formalization targets
The goal is §4.4 Theorem 1 (Gauss–Markov): with $K_0 = (W^\top Q^{-1} W)^{-1} W^\top Q^{-1}$,
$$K_0 W = I, \qquad E\big[(K_0 y - \beta)_i^2\big] \le E\big[(K y - \beta)_i^2\big] \quad \text{for every } i \text{ and every } K \text{ with } KW = I,$$
with error covariance
$$E\big[(K_0 y - \beta)(K_0 y - \beta)^\top\big] = (W^\top Q^{-1} W)^{-1}.$$
Milestones: the deterministic least-squares estimate $\hat\beta = (W^\top W)^{-1} W^\top y$ (§4.3 Theorem 1); the book's deterministic reduction — minimize the diagonal entries of $KQK^\top$ subject to $KW = I$ (p. 85); the minimum-variance estimate $\hat\beta = E[\beta y^\top] (E[y y^\top])^{-1} y$ for random $\beta$ (§4.5 Theorem 1); and the information-form identities $RW^\top(WRW^\top + Q)^{-1} = (W^\top Q^{-1}W + R^{-1})^{-1}W^\top Q^{-1}$ and $R - RW^\top(WRW^\top+Q)^{-1}WR = (W^\top Q^{-1}W + R^{-1})^{-1}$ (§4.5 Corollary 2).
## Significance
The Gauss–Markov theorem justifies weighted least squares as the optimal linear unbiased procedure and is the standard benchmark against which biased and nonlinear estimators are measured. The minimum-variance estimate of §4.5 is the Bayesian counterpart with prior covariance $R$; the information-form identities connect the two and exhibit Gauss–Markov as the limit $R^{-1} \to 0$. Mission III builds the recursive (Kalman) estimator directly on these results.
All results are classical and proved in the source. Mathlib has mature measure-theoretic integration but, to date, no Gauss–Markov theorem and no linear estimation theory; the matrix milestones (trace reduction, information form) are also absent as stated. The probabilistic statements here are deliberately phrased with elementary integrals of products of real-valued components — no Bochner integration of vector-valued maps — so they are approachable with `MeasureTheory.integral` alone.
## Difficulty
The subtlety is bookkeeping, not depth. Unbiasedness must be encoded as the algebraic constraint $KW = I$ (the book proves the equivalence with $E\hat\beta = \beta$ for all $\beta$); the componentwise variance claim is strictly stronger than the trace claim and requires the per-component minimum norm argument, not a single matrix inequality. Positive definiteness of $Q$ enters through invertibility of $W^\top Q^{-1} W$, which itself needs the linear independence of the columns of $W$ — dropping either hypothesis makes the goal false. In the probabilistic statements every integral needs an integrability hypothesis; the drafts supply integrability of all pairwise products of components, from which integrability of every derived expression follows.
## Formalization scope
Random vectors are plain functions `Ω → Fin m → ℝ` on a `MeasurableSpace Ω` with a probability measure `μ`; second moments are hypotheses of the form `∫ ω, ε ω i * ε ω j ∂μ = Q i j` with explicit `Integrable` assumptions; no independence, Gaussianity, or distributional assumptions are used anywhere. Matrices are `Matrix (Fin m) (Fin n) ℝ` with Mathlib's `Matrix.PosDef`, nonconstructive inverse `⁻¹`, and `mulVec`. Norms on parameter space are written as explicit finite sums of squares, avoiding any ambiguity between Euclidean and supremum norms on pi types. The estimators under comparison are strictly linear ($\hat\beta = Ky$, no affine offset), exactly as in the source; §4.5's affine extension (its Problem 6) is out of scope.
## Selected references
- David G. Luenberger, *Optimization by Vector Space Methods*, John Wiley & Sons, 1969. Chapter 4, pp. 78–102. ISBN 0-471-55359-X.
- A. C. Aitken, *On least squares and linear combination of observations*, Proc. Roy. Soc. Edinburgh 55 (1935), 42–48 (the weighted-least-squares form of Gauss–Markov).
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Vector Space Methods I: Minimum Norm Problems in Hilbert SpaceTextbook
## Motivation
Luenberger's *Optimization by Vector Space Methods* (Wiley, 1969) organizes a large part of optimization theory around a single geometric idea: minimum norm problems in inner product spaces, solved by orthogonal projection. Chapter 3 is the technical heart of that program. Its projection theorem and normal equations underlie least-squares data fitting, Fourier approximation, minimum-energy control, and the whole statistical estimation theory of Chapter 4 — which Missions II and III of this series formalize on top of the present one.
## Setting
Throughout, spaces are **real**. A **pre-Hilbert space** is a real vector space $X$ with an inner product $\langle\cdot,\cdot\rangle$ inducing the norm $\|x\| = \langle x,x\rangle^{1/2}$; a **Hilbert space** $H$ is a complete pre-Hilbert space. Vectors $x, y$ are **orthogonal** when $\langle x, y\rangle = 0$; for a subset $S$, the **orthogonal complement** $S^\perp$ is the set of vectors orthogonal to every element of $S$. Given $y_1,\dots,y_n \in H$, their **Gram matrix** is $G(y_1,\dots,y_n)_{ij} = \langle y_i, y_j\rangle$ and its determinant $g(y_1,\dots,y_n)$ is the **Gram determinant**. A **linear variety** is a translate $x + M$ of a subspace $M$.
## Formalization targets
The goal is §3.10 Theorem 2, the **dual approximation problem**: for linearly independent $y_1,\dots,y_n \in H$ and constants $c_1,\dots,c_n$, among all $x \in H$ satisfying the constraints
$$\langle x, y_i\rangle = c_i, \qquad i = 1,\dots,n,$$
there is a unique vector of minimum norm, and it has the form
$$x_0 = \sum_{i=1}^n \beta_i\, y_i, \qquad \text{where} \qquad \sum_{j=1}^n \beta_j \langle y_j, y_i\rangle = c_i .$$
The milestone list follows the chapter's own development: the projection theorem in its pre-Hilbert form (§3.3 Theorem 1) and classical form (§3.3 Theorem 2), the orthogonal decomposition $H = M \oplus M^\perp$ with $M^{\perp\perp} = M$ (§3.4 Theorem 1), the normal equations and Gram matrices (§3.6), the Gram determinant formula $\delta^2 = g(y_1,\dots,y_n,x)/g(y_1,\dots,y_n)$ for the minimum distance (§3.6 Theorem 1), best approximation by Fourier sums over orthonormal families (§3.7, §3.9), minimum norm over a linear variety (§3.10 Theorem 1), and the extension from subspaces to closed convex sets with its variational inequality characterization (§3.12 Theorem 1).
## Significance
The dual approximation theorem converts an infinite-dimensional constrained minimum norm problem into an $n \times n$ linear system — the book's model example of finite reduction, applied there to minimum-energy control of a motor (§3.11) and, in Chapter 4, to every linear estimation problem: least squares, Gauss–Markov, and recursive (Kalman) estimation are all instances of these results in a Hilbert space of random variables.
All results here are classical and proved in the source; the mission's product is a faithful machine-checked development with reusable statements. Mathlib already contains close relatives of several milestones (orthogonal projection onto complete subspaces, `Submodule.orthogonal`), so part of the work is connecting the book's formulations to that library; the Gram determinant distance formula and the dual approximation theorem itself have no direct Mathlib counterpart.
## Difficulty
The individual milestones are standard Hilbert space theory. The care is in the statements, not tricks: the pre-Hilbert version of the projection theorem asserts uniqueness and the orthogonality characterization *without* existence, while existence requires completeness and closedness — conflating the two versions produces unprovable or vacuous statements. The Gram determinant formula requires the $(n+1) \times (n+1)$ Gram matrix of the extended family $(y_1,\dots,y_n,x)$, where index bookkeeping (`Fin.snoc`) is easy to get wrong. In §3.12 the variational inequality $\langle x - k_0, k - k_0\rangle \le 0$ replaces the equality characterization valid for subspaces; the inequality direction is a known trap.
## Formalization scope
The development commits to: real scalars (the book allows complex; this series does not), an abstract space `H : Type` with `[NormedAddCommGroup H] [InnerProductSpace ℝ H]` and `[CompleteSpace H]` exactly where the source assumes a Hilbert space; subspaces as `Submodule ℝ H` with explicit `IsClosed` hypotheses; finite families as `Fin n → H`; Gram matrices as `Matrix (Fin n) (Fin n) ℝ` via `Matrix.of`; minimum distances as infima (`⨅`) over coerced submodules. Best approximation statements are phrased as explicit inequalities `‖x - m₀‖ ≤ ‖x - m‖` rather than through any projection operator, so they are usable without choosing Mathlib's `orthogonalProjection` API. Statements deliberately carry no more hypotheses than the source: §3.3 Theorem 1 and the normal equations hold in any real inner product space; completeness appears only where existence is claimed.
Proofs are expected to lean on Mathlib's inner product space library; contributions of reusable bridging lemmas (e.g. between `⨅`-formulations and `orthogonalProjection`) are welcome as child lemmas via proof sketches.
## Selected references
- David G. Luenberger, *Optimization by Vector Space Methods*, John Wiley & Sons, 1969. Chapter 3, pp. 46–77. ISBN 0-471-55359-X.